mm F G r PHYS:1200 LECTURE 3 MECHANICS (2)

Transcription

1 1 PHYS:1200 LECTURE 3 MECHANICS (2) In Lecture 2 we learned that an object will continue in its state of motion unless something acts to change that state. Forces acting on an object can change its state of motion. In Lecture 3 we will consider one of the most common forces that changes the state of motion of objects gravity. We are all subject to the force of gravity that is exerted on us by the earth, so it is really part of our everyday experience. We throw an object upward and it falls back to the ground this is the effect of the force of gravity acting while the object is going up and coming back down. We will return to this example a little later Newton s Universal Law of Gravity. First we give some of the basic facts of the law of gravity which was discovered by Newton in Gravity is the force that keeps us on the earth, the moon in its orbit around the earth, and the earth (and all the other planets) revolving around the sun. In fact, gravity holds the entire universe together. Our weight is simply the force of gravity on us by the earth the earth is always pulling us downward (actually toward its center). Gravity is a fundamental force of nature and is an example of what is called a non contact force. You do not have to be in contact (touching) with the earth in order for the force to act on us. When we jump us, gravity acts to pull us down. Gravity is an attractive (tries to pull them together) force that acts between any two objects (not just between us and the earth) and it depends on the masses of the two objects. The greater the masses, the greater the force. Gravity also depends on how close the objects are to each other the closer the objects are the greater the force. Newton s law of gravity for 2 masses separated by a distance r is given by: Newton s Law of Universal Gravitation mm F G r 1 2, 2 m1 r m2

2 2 where G is a constant. This formula says that the force between the two masses (which is attractive, i.e., tries to pull them toward each other) depends on the product of the two masses divided by the square of their separation r. You and the person sitting next to you are attracted toward each other by the force of gravity you exert an attractive force on the person next to you and the person next to you exerts an attractive force on you. Now you will immediately comment that I do not feel an attractive force exerted on me by the person sitting next to me. We are not aware of this force because it is a very, very small force and it is greatly overwhelmed by all the other forces acting on you like the friction between you and your seat. The force is small because the masses involved are relatively small (typically kg). On the other hand, we definitely feel the force of the earth s gravity on us (our weight) because the mass of the earth is huge (6x10 24 kg). We will now explore the effects of gravity on the objects near the earth s surface The Acceleration Due to Gravity. When an object falls to earth it is acted on by two forces: gravity and a drag force (air resistance) due to the air molecules hitting it. For objects that fall to earth from a distance of several feet or meters, the effect of air resistance can usually be neglected. To simplify the discussion of gravity, we will just ignore air resistance for now. When an object is dropped it falls to the ground this is what is called free fall. If a video of the motion of the object is made and analyzed frame by frame we will observe that as it falls it does not maintain a constant speed but its speed increases as it falls the closer it is to the ground the higher its speed. This is an example of what we call accelerated motion. Acceleration means increase in velocity and it is a measure of the rate at which the velocity increases it is the amount of change in velocity divided by the time interval over which the change occurs. In mathematical terms we write for the acceleration a: Definition of acceleration Δv a = Δt [1] Δv is read delta v and means the change in velocity, and Δt read delta t is the time interval. This is just the usual meaning of acceleration when your car accelerates, its speed increases; you push down on the accelerator to go faster.

3 3 Example 3.1: suppose an object starts from rest ( v = 0) and then is accelerated to a speed of 10 m/s (meters per second) over a time interval of 5 s (seconds); its acceleration is then a = 10m/s 5s = 2m / s per s, or 2m / s 2, or, 2 meters per second squared. Note that an object that moves with a constant speed has an acceleration of zero. Now that we have discussed the general concept of acceleration, we can get back to the specific case of the acceleration due to gravity. If we analyze in detail the video of the falling object (see slide #10) we find that each second that it falls, its speed increases by approximately 10 m/s (the precise number is 9.8, but in this course we will use 10 to keep the arithmetic simple). This is illustrated by the Table in slide #11. Since the speed increases 10 m/s each second, the acceleration is 10 m/s/s = 10 m/s 2. This important parameter is called the acceleration due to gravity and is given the special symbol g, so that Acceleration due to gravity g = 9.8ms 10ms 32 fts. [2] a. Mass and weight. We can now discuss how to calculate the weight of an object, which recall is just the force of the earth s gravity on it. First of all, we need to emphasize that weight and mass are not the same! Weight is a FORCE and mass is a measure of how much matter is in the object in everyday language they are apples and oranges. The mass of an object, which we agreed to measure in kilograms (kg), is a parameter that would be the same regardless of where the object were in the universe. On the other hand, the weight of an object is the force exerted on it by whatever large planet or other celestial body it happens to be near. We have all heard that your weight is less on the moon than on the earth. This is because the mass of the moon is much less than the mass of the earth. (The acceleration due to moon s gravity is 1/6 the acceleration on earth.) An object having a mass m on earth has a weight w calculated by multiplying m times g: weight w = m g, or simply m g. [3]

4 4 Example 3.2: the weight of an object of mass m = 50 kg is w = 50 kg x 10 m/s 2 = 500 kg m/s 2. The units for weight is the combination of the mass unit multiplied by the acceleration unit or kg m/s 2. Since weight is just a specific example of a force, kg m/s 2 is the general unit for any force in our scientific system of units. Sometimes, a unit formed by the combination of other units is given a special name; in this case 1 kg m/s 2 is given the name 1 Newton (N), in honor of Sir Isaac Newton. So in the above example, the weight of the 50 kg object is 500 N. In the system of units used in the US (called the English system, even though it is not used anymore in the UK) (Liberia and Burma still use the English system also) weight is measured in the British force unit of pounds (lb), with the conversion that 1 N = lb. Often, a person s weight might be quoted as the equivalent mass in kg. The reason for using kg for weight is that even though mass and weight are not the same, your weight is uniquely determined by your mass, so you can compare weights by comparing masses if object B has twice the mass as object A, it also weights twice as much. A mass of 1 kg has a weight of mg (1 kg) (10 m/s 2 ) = 10 N. Using the conversion 1 N = lb, 10 N 2.2 lb, so we say that 2.2 lb is approximately equivalent to 1 kg. (the symbol means approximately equal to) Example 3.3: What is the weight in kg of a 150 lb person? Solution 1kg w 150 lb 68 kg. Note that I put weight in quotation marks because kg is a 2.2 lb unit of mass and weight is a unit of force. What is meant here is that a 150 lb person has an equivalent mass of 68 kg. A 68 kg mass has a weight of approximately 680 N. How is weight measured? On a scale what is a scale? If you hang an object from a spring it stretches the spring. So the amount that the spring is stretched can be calibrated so that it gives a weight measurement. Springs are interesting objects in themselves which we will discuss in a later lecture. b. Galileo s experiments on free fall. Galileo performed the first experiments on gravity, supposedly by dropping various objects from the Leaning Tower of Pisa, although historians are still arguing about whether or not this is truth or legend. What is clear is that wherever he did

5 5 the experiments, he came to the correct conclusions about free fall. Now to fully appreciate Galileo s contributions to the science of motion, we need to realize that the clock as we know it was not yet invented in his time. The fact that he was able to get the physics correct without an instrument that accurately measured time is quite remarkable. Galileo was an ingenious fellow! How did he measure time? Galileo was trained as a physician and he knew that the heart beat (pulse) was a regular phenomenon so he could use his own pulse as a stop watch. (A normal male has a heart rate of beats per minute.) He also realized that an object tied to the end of a string (pendulum) would swing back and forth at regular intervals of time, and the shorter the string, the shorter the time. So each swing of the pendulum is a tick of a clock. Galileo dropped various objects (different masses) from various heights and measured how long it took them to reach the ground. The conclusions from his experiment were: GALILEO: All objects, regardless of their mass, and in the absence of air resistance, fall to earth with the same acceleration g = 10 m/s 2. This will be shown in class by a demonstration in which a feather and a 25 cent coin fall in a glass cylinder in which the air was pumped out, thus essentially eliminating air resistance. (This experiment was also performed on the moon (where there is no air) by Cmdr. David Scott on the Apollo 15 mission.) To obtain quantitative results from his free fall experiments, Galileo made us of inclined planes (slide # 19). He realized that an object sliding down an inclined plane was still falling under the influence of gravity, but the effect of gravity would be reduced depending on the angle of the plane relative to the horizontal. (To reduce the effect of friction between the object and the plane, Galileo used spherical objects that rolled down the plane nearly frictionlessly. We can perform experiments today using the air track that is tilted.) For example, if the plane of the incline is tilted relative to the horizontal by 30 degrees, the effect of gravity is reduced by a factor of 2, that is the effective g = g/2 = 5 m/s 2. The advantage of using the inclined plane was that the time for the object to reach the bottom of the plane was longer than the case in which the object fell straight down from the same height as the top of the plane. This meant that the time was longer and therefore could be measured more accurately. From these experiments, Galileo

6 6 was able to obtain a mathematical relation for the distance that an object fell as a function of time. This will be discussed in the next lecture. c. Up and down: the effect of gravity. You toss a ball straight up, it continues to move upward until it reaches its highest point, then falls back down. What is the effect of gravity as the ball rises and then as it falls back down? First, we must realize that the force of gravity on an object near the earth (i.e., its weight) always acts in the downward direction (technically toward the center of the earth, but locally we think of the earth as flat so the force is down.) When you toss a ball upward, you give it some initial upward speed. The downward force of gravity then acts to slow the ball down (reducing its speed) as it rises, until it reaches its highest point where it is instantaneously at rest, i.e., v = 0 at the very top of the ball s path. Gravity then continues to pull the ball downward from its highest point, and as it falls its speed increases. Keep in mind that weight is a force and forces cause changes in velocity. On the way up the velocity decreases, on the way down velocity increases, but the force (weight) always acts downward.

Chapter 3: Falling Objects and Projectile Motion 1. Neglecting friction, if a Cadillac and Volkswagen start rolling down a hill together, the heavier Cadillac will get to the bottom A. before the Volkswagen.

Practice Midterm 1 1) When a parachutist jumps from an airplane, he eventually reaches a constant speed, called the terminal velocity. This means that A) the acceleration is equal to g. B) the force of

Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled

Supplemental Questions The fastest of all fishes is the sailfish. If a sailfish accelerates at a rate of 14 (km/hr)/sec [fwd] for 4.7 s from its initial velocity of 42 km/h [fwd], what is its final velocity?

5. Forces and Motion-I 1 Force is an interaction that causes the acceleration of a body. A vector quantity. Newton's First Law: Consider a body on which no net force acts. If the body is at rest, it will

At the skate park on the ramp 1 On the ramp When a cart rolls down a ramp, it begins at rest, but starts moving downward upon release covers more distance each second When a cart rolls up a ramp, it rises

1. What is the average speed of an object that travels 6.00 meters north in 2.00 seconds and then travels 3.00 meters east in 1.00 second? 9.00 m/s 3.00 m/s 0.333 m/s 4.24 m/s 2. What is the distance traveled

VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how

How Rockets Work Whether flying a small model rocket or launching a giant cargo rocket to Mars, the principles of how rockets work are exactly the same. Understanding and applying these principles means

* By request, but I m not vouching for these since I didn t write them Exam 2 is at 7 pm tomorrow Conflict is at 5:15 pm in 151 Loomis There are extra office hours today & tomorrow Lots of practice exams

Lesson 29: Newton's Law of Universal Gravitation Let's say we start with the classic apple on the head version of Newton's work. Newton started with the idea that since the Earth is pulling on the apple,

Chapter 10 Energy and Work 10.1 Quantitative 1) A child does 350 J of work while pulling a box from the ground up to his tree house with a rope. The tree house is 4.8 m above the ground. What is the mass

Standard 7.3.17: Investigate that an unbalanced force, acting on an object, changes its speed or path of motion or both, and know that if the force always acts toward the same center as the object moves,

Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

Candidates should be able to : Derive the equations of motion for constant acceleration in a straight line from a velocity-time graph. Select and use the equations of motion for constant acceleration in

This booklet will discuss some of the principles involved in the design of a roller coaster. It is intended for the middle or high school teacher. Physics students may find the information helpful as well.

CT16-1 Which of the following is necessary to make an object oscillate? i. a stable equilibrium ii. little or no friction iii. a disturbance A: i only B: ii only C: iii only D: i and iii E: All three Answer:

Exam Three Momentum Concept Questions Isolated Systems 4. A car accelerates from rest. In doing so the absolute value of the car's momentum changes by a certain amount and that of the Earth changes by:

Section 4: The Basics of Satellite Orbits MOTION IN SPACE VS. MOTION IN THE ATMOSPHERE The motion of objects in the atmosphere differs in three important ways from the motion of objects in space. First,

The Big Idea Acceleration is caused by force. All forces come in pairs because they arise in the interaction of two objects you can t hit without being hit back! The more force applied, the greater the

Work by Integration 1. Finding the work required to stretch a spring 2. Finding the work required to wind a wire around a drum 3. Finding the work required to pump liquid from a tank 4. Finding the work

More Chapter 3 Projectile motion simulator http://www.walter-fendt.de/ph11e/projectile.htm The equations of motion for constant acceleration from chapter 2 are valid separately for both motion in the x

A cannon shoots a clown directly upward with a speed of 20 m/s. What height will the clown reach? How much time will the clown spend in the air? Projectile Motion 1:Horizontally Launched Projectiles Two

CTN-12. Consider a person standing in an elevator that is moving upward at constant speed. The magnitude of the upward normal force, N, exerted by the elevator floor on the person's feet is (larger than/same

XX. Introductory Physics, High School High School Introductory Physics Test The spring 2013 high school Introductory Physics test was based on learning standards in the Physics content strand of the Massachusetts

Exam Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) A person on a sled coasts down a hill and then goes over a slight rise with speed 2.7 m/s.

After completing this chapter you should be able to solve problems involving motion in a straight line with constant acceleration model an object moving vertically under gravity understand distance time

4277(a) Semester 2, 2011 Page 1 of 9 THE UNIVERSITY OF SYDNEY EDUH 1017 - SPORTS MECHANICS NOVEMBER 2011 Time allowed: TWO Hours Total marks: 90 MARKS INSTRUCTIONS All questions are to be answered. Use

Motion Graphs It is said that a picture is worth a thousand words. The same can be said for a graph. Once you learn to read the graphs of the motion of objects, you can tell at a glance if the object in

Lesson 3: Energy, Momentum, and Understanding Car Crashes Many of us have lost students to violent motor vehicle crashes. In the United States, motor vehicle crashes are the number one cause of death among

In the previous section, you have come across many examples of motion. You have learnt that to describe the motion of an object we must know its position at different points of time. The position of an

1 DIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION Daniel S. Orton email: dsorton1@gmail.com Abstract: There are many longstanding

Dynamics of Iain M. Banks Orbitals Richard Kennaway 12 October 2005 Note This is a draft in progress, and as such may contain errors. Please do not cite this without permission. 1 The problem An Orbital

PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be

Viscosity It is the property of a liquid due to which it flows in the form of layers and each layer opposes the motion of its adjacent layer. Cause of viscosity Consider two neighboring liquid layers A

Science teaching unit Disclaimer The Department for Children, Schools and Families wishes to make it clear that the Department and its agents accept no responsibility for the actual content of any materials

Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely

Student Activity Sheet 6 Page 1 Name physics, technology and engineering in automobile racing review/assessment questions 1. Draw a free-body diagram for a block being pushed across the floor. 2. Use all

Recitation Week 4 Chapter 5 Problem 5.5. A bag of cement whose weight is hangs in equilibrium from three wires shown in igure P5.4. wo of the wires make angles θ = 60.0 and θ = 40.0 with the horizontal.

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,

Chapter 7 Momentum and Impulse Collisions! How can we describe the change in velocities of colliding football players, or balls colliding with bats?! How does a strong force applied for a very short time

Physics 121 Homework Problems, Spring 2014 1-1. Write out your solution to all parts of this problem neatly on a piece of 8.5 11-inch paper and turn it in at the slotted boxes across the hallway from N373

Provided by TryEngineering - Lesson Focus This lesson focuses on parachute design. Teams of students construct parachutes from everyday materials. They then test their parachutes to determine whether they

2.8 Functions and Mathematical Models 131 2.8 FUNCTIONS AND MATHEMATICAL MODELS At one time Conway would be making constant appeals to give him a year, and he would immediately respond with the date of

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

2 ONE- DIMENSIONAL MOTION Chapter 2 One-Dimensional Motion Objectives After studying this chapter you should be able to derive and use formulae involving constant acceleration; be able to understand the

Chapter 9 9. Figure 9-36 shows a three particle system. What are (a) the x coordinate and (b) the y coordinate of the center of mass of the three particle system. (c) What happens to the center of mass

The Science of Archery Godai Katsunaga Purpose To provide insight into the physics of arrow flight and show how archers adapt their equipment to maximize effectiveness. Archery Archery is one of the events

5 Question 1. [Marks 28] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding

Force & Motion Activity Tub Designed to meet these objectives: Students will be able to describe Newton s First, Second, and Third Laws of Motion and identify examples of these laws at work in the world

Lesson 11 Physics 168 1 Oscillations and Waves 2 Simple harmonic motion If an object vibrates or oscillates back and forth over same path each cycle taking same amount of time motion is called periodic

150 hapter 4 Forces and Motion I: Newton s Laws Questions and Problems In a few problems, you are given more data than you actually need; in a few other problems, you are required to supply data from your

6 WORK and ENERGY Chapter 6 Work and Energy Objectives After studying this chapter you should be able to calculate work done by a force; be able to calculate kinetic energy; be able to calculate power;